String bordism

1 Introduction

$String$$== Introduction == <wikitex>; $String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations $$ \xymatrix{BO\!\left< 8 \right>\ar[d] & \ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \ BO\ar[r]^{w_1}& K({\mathbb Z}/2,1) } $$ In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below. There are various models for the String group. </wikitex> == Generators == <wikitex>; * For $i \le1 6$ the natural map $\Omega_i^{fr} \cong \Omega_i^{String}$ is an isomorphism. See [[Framed bordism|framed bordism]] for these groups. The following groups come from \cite{Giambalvo1971|p. 538}. * $\Omega_7^{String} = 0$. * $\Omega_8^{String} \cong \Zz \oplus \Zz/2$ generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}. * $\Omega_9^{String} \cong (\Zz/2)^2$ generated by [[Exotic spheres|exotic 9-spheres]]. * $\Omega_{10}^{String} \cong \Zz/2$ generated by an [[Exotic spheres|exotic 10-sphere]]. * $\Omega_{11}^{String} = 0$. * $\Omega_{12}^{String} \cong \Zz$. * $\Omega_{13}^{String} = 0$. * $\Omega_{14}^{String} \cong \Zz/2$ generated by $S^7 \times S^7$ with the analogue of a Lie invariant framing. * $\Omega_{15}^{String} \cong \Zz/2$ genreated by the [[Exotic spheres|exotic 15-sphere]]. * $\Omega_{16}^{String} \cong \Zz^2$. </wikitex> == Additive structure == <wikitex>; The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971} also for the first 16 bordism groups.) </wikitex> == The Witten genus == <wikitex>; At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.) </wikitex> ==Characteristic numbers== <wikitex>; The Witten genus can be refined to a map of structured ring spectra $$W: MString \longrightarrow TMF$$ from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms. This spectrum was developed by Mike Hopkins (see \cite{Hopkins2002}.) It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.) </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}String$-bordism or $O\!\left< 8 \right>$$O\!\left< 8 \right>$-bordism is a special case of a B-bordism. It comes from the tower of fibrations

$\displaystyle \xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}& K({\mathbb Z}/2,1) }$

In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$$BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$$BSpin$ given by half of the first Pontryagin class. The name $String$$String$-group is due to Haynes Miller and will be explained below. There are various models for the String group.

2 Generators

• For $i \le1 6$$i \le1 6$ the natural map $\Omega_i^{fr} \cong \Omega_i^{String}$$\Omega_i^{fr} \cong \Omega_i^{String}$ is an isomorphism. See framed bordism for these groups. The following groups come from [Giambalvo1971, p. 538].
• $\Omega_7^{String} = 0$$\Omega_7^{String} = 0$.
• $\Omega_8^{String} \cong \Zz \oplus \Zz/2$$\Omega_8^{String} \cong \Zz \oplus \Zz/2$ generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
• $\Omega_9^{String} \cong (\Zz/2)^2$$\Omega_9^{String} \cong (\Zz/2)^2$ generated by exotic 9-spheres.
• $\Omega_{10}^{String} \cong \Zz/2$$\Omega_{10}^{String} \cong \Zz/2$ generated by an exotic 10-sphere.
• $\Omega_{11}^{String} = 0$$\Omega_{11}^{String} = 0$.
• $\Omega_{12}^{String} \cong \Zz$$\Omega_{12}^{String} \cong \Zz$.
• $\Omega_{13}^{String} = 0$$\Omega_{13}^{String} = 0$.
• $\Omega_{14}^{String} \cong \Zz/2$$\Omega_{14}^{String} \cong \Zz/2$ generated by $S^7 \times S^7$$S^7 \times S^7$ with the analogue of a Lie invariant framing.
• $\Omega_{15}^{String} \cong \Zz/2$$\Omega_{15}^{String} \cong \Zz/2$ genreated by the exotic 15-sphere.
• $\Omega_{16}^{String} \cong \Zz^2$$\Omega_{16}^{String} \cong \Zz^2$.

3 Additive structure

The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\!\left< 8 \right>$$BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the framed bordism groups. It is known that only 2 and 3 torsion appears (see [Giambalvo1971] also for the first 16 bordism groups.)

4 The Witten genus

At the end of the 80s Ed Witten were studying the $S^1$$S^1$-equivariant index of the Dirac operator on a loop space of a $4k$$4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$$LM$ this would mean that $M$$M$ is $String$$String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$$k$. Nowadays this is called the Witten genus (see [Segal1988].)

5 Characteristic numbers

The Witten genus can be refined to a map of structured ring spectra

$\displaystyle W: MString \longrightarrow TMF$ $$\displaystyle W: MString \longrightarrow TMF$$

from the Thom spectrum of String bordism to the spectrum $TMF$$TMF$ of topological modular forms. This spectrum was developed by Mike Hopkins (see [Hopkins2002].) It is supposed to play the same role for $String$$String$-bordism as $KO$$KO$-theory does for $Spin$$Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$$W$ gives characteristic numbers which together with $KO$$KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$$String$ bordism class. Moreover, $tmf$$tmf$ is supposed to be a direct summand of $MString$$MString$ as the orientation map $W$$W$ is shown to be surjective in homotopy (see [Hopkins&Mahowald2002].)

6 References

• [Giambalvo1971] V. Giambalvo, On $\langle 8\rangle$$\langle 8\rangle$-cobordism, Illinois J. Math. 15 (1971), 533–541. MR0287553 (44 #4757) Zbl 0221.57019
• [Hopkins&Mahowald2002] M. Mahowald and M. Hopkins, The structure of 24 dimensional manifolds having normal bundles which lift to $BO[8]$$BO[8]$, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., 293 (2002), 89–110. MR1887530 (2003b:55007) Zbl 1012.57041
• [Hopkins2002] M. J. Hopkins, Algebraic topology and modular forms, (2002), 291–317. MR1989190 (2004g:11032) Zbl 1031.55007
• [Laures2004] G. Laures, $K(1)$$K(1)$-local topological modular forms, Invent. Math. 157 (2004), no.2, 371–403. MR2076927 (2005h:55003) Zbl 1078.55010
• [Segal1988] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Séminaire Bourbaki, Vol. 1987/88, Astérisque No. 161-162 (1988), Exp. No. 695, 4, (1989) 187–201. MR992209 (91b:55005) Zbl 0686.55003

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